The theory of modular forms is an important tool of contemporary mathematics, in particular in number theory and arithmetic geometry of elliptic curves (see for instance the spectacular proof of Fermat’s last theorem). The books devoted to this part of mathematics are characterized by deeply developed theories seemingly far from concrete computations. This book is highly non-traditional in this respect. Besides developing the theory of modular forms, the author brings the reader down to earth by addressing the computational aspects of the theory. Various theoretical aspects are transformed into algorithms and practical programs. To formulate them, the author uses his free open source computer algebra system SAGE (Software for Algebra and Geometry Experimentation).
The main body of the book is formed of eleven chapters, complemented by an appendix “Computing in Higher Rank” of about 50 pages and written by P. E. Gunnells. Each chapter contains numerous examples and exercises. The solutions to most of the exercises can be found in the final chapter. The introductory first chapter starts with basic material about congruence subgroups and modular forms on the upper-half plane, and lists some main applications of modular forms throughout mathematics. The remaining chapters then cover a variety of important aspects of the theory. The exposition is very clear and vivid showing the author’s mastery in the subject. It contains illustrating examples, many methodological comments and references to the relevant research literature for further reading. Since the prerequisites are moderate, the highly original approach makes the book suitable for advanced undergraduates in related algebraic number theory or arithmetic geometry, and the wealth of explicit algorithms and the unique approach makes the book almost indispensable for mathematicians interested in the field.